Michel Bonnefont

On gradient bounds for the heat kernel on the Heisenberg group

It is known that the couple formed by the two dimensional Brownian motion and its Levy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel.

Intertwining Relations for One-Dimensional Diffusions and Application to Functional Inequalities

Following the recent work Chafaï and Joulin (Bernoulli 19:1855–1879, 2013) fulfilled in the discrete case, we provide in this paper new intertwining relations for semigroups of one-dimensional diffusions. Various applications of these results are investigated, among them the famous variational formula of the spectral gap derived by Chen and Wang (Trans. Am. Math. Soc. 349:1239–1267, 1997) together with a new criterion ensuring that the logarithmic Sobolev inequality holds. We complete this work by revisiting some classical examples, for which new estimates on the optimal constants are derived.